摄像机线性标定方法
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摘要
摄像机标定作为计算机视觉中的基本问题。在实际应用中,精度高、鲁棒性强的方法才会被大家普遍运用,而且方法的简便性也是人们关心的一个方面。作为标定基元,由于确定一条直线比一个点需要更多的参数,因此,基于直线的标定方法一般来说要比基于点的标定方法具有更高的鲁棒性。同理,基于二次曲线的标定方法的鲁棒性也就更高。但是,基元的次数越高,方程求解也就越困难。线性方法计算简单,在数值上比非线性方法稳定得多。人们一直在寻找基于高次的标定基元的线性标定方法,这方面也有不少的成果。本文首先列举了其中的一些方法,并指出了某些方法的不足。最后,给出了一种基于任意分布的共面圆的摄像机线性标定方法。该方法不需要对模板中的二次曲线进行定位,只要三个(或更多)圆,这些圆可以是两两同心、相交、相切或是分离的(事实上,我们的算法可以通过两圆的像信息反求出两圆的位置关系);不需要模板与图像之间的匹配;也无需求解非线性方程组。本文从几何角度对算法进行形象描述,并从代数的角度给出了严格论证。模拟和真实图像实验表明,该算法精确高,鲁棒性强。由于无需知道圆的任何几何信息,也不用模板与图像的匹配信息,因而具有较好的实用性。
The motivation to study the geometry if conics arises from the fact that conics have more geometric information,and can be more robustly than points and lines.The the camera calibration method proposed in this paper,which based on the arbitrary distributed three(or more) coplanar circles,can deal with all the distributions of each two circles such as intersection,tangent, including,separation.What's more,we shouldn't know any of the properties of circles,and needn't handling matching.
As the absolute conic has all the information needed for the camera's intrinsic parameters,we now just need to find the image of the absolute conic. Based on this,many people have proposed some calibration methods.[27][2] give the calibration using a circle and its diameters;the linear method introduced by[6]based on two rectangulars;which the calibration algorithms in [7][14]based are only circles,but the method is not linear.
Aimed at computing the image of the absolute conic,which is also a conic,we should find at least five points which lay in the conic.Fortunately, the generally called circles p0ints are just such points.So the problem is now that how to find the images of at least three pairs of circle points.Realizing that circle points are the intersection of a circle and the infinite line,our task comes to finding the intersection of the images of a circle and the infinite line which is called the vanishing line.We can easily get the image of a circle which lav on the model plane,the coming difficulty is how to find the vanishing line.
Suppose there are three circles which are distributed arbitrarily,we will show how to find the vanishing line by using these there circles.We should at first group them so that two circles are in a group,then the image of at least one infinite point(which lay in the infinite line) will be computed from the image of each group.So using the there image points lay in the vanishing line,we can be fitted(of course we can use more than three circles in the model plane so as to get more image points of the vanishing line to fit it).
Suppose there is a point X=(x,y,t)~T in the projective plane of the circles O_1,O_2,the necessary and sufficient condition of X being the common pole is that O_2X=λO_1X,that is, (O_2-λO_1)X=0,(0-2-1) withλ∈R\{0}.
We have pointed out the fact that:the common poles of O_1,O_2 must include at least one infinite point.
Under the projective transformation H,circles O_1,O_2 have the image of quadratic curves C_1=λ_1H~(-T)O_1H~(-1),C_2=λ_2H~(-T)O_2H~(-1),withλ_1,λ_2 R\{0}.Substitute this into(0-2-1),we have:(C_2-λ(λ_2/λ_1)C_1)HX=0,that is, (C_2-λ′C_1)HX=0.(0-2-2) Therefore,HX is the common pole of C_1 and C_2.Because H is nonsingular, andλ_1,λ_2 are non-zero,so equation(0-2-1) and(0-2-2) are equivalent. This is just the algebra description of the invariance of pole and polar line about projective transformation.
Therefore,the common poles of C_1,C_2 which are the images of the two circle must include at least one infinite point.
The following is the method of finding the vanishing line base on the equation(0-2-3). det(C_2-λ′C_1)=0.(0-2-3)
proposition 0.1
(1) If the equation(0-2-3) haven't repeated root(in this situation the preimage of C_1,C_2 are neither concentric nor tangent circles),compute the independent real rootλ′,then substituteλ′into equation(0-2-2),the eigenvector HX is the image of infinite point d_(m∞).
(2) If equation(0-2-3) has repeated root,then it must be twofold root,compute this twofold rootλ′and substitute it into the matrix C_2-λ′C_1:
(a) If the matrix C_2-λ′C_1 rank of 1(showing that the preimage of C_1,C_2 are concentric circles),substitute thisλ′into equation(0-2-2),the two independent eigenvectors HX~((1)) and HX~2 are the image of infinite points d_(m∞)~((1)) and d_(m∞)~((2));
(b) If the matrix C_2-λ′C_1 rank of 2(showing that the preimage of C_1,C_2 are tangent circles),compute the independent real root of equation(0-2-3),then substitute it into equation(0-2-2),and the eigenvector HX is the image of infinite point d_(m∞).
Till now,we have find the image of a infinite point.
If there are three circles in the model plane,they can be divided into three groups,and at least one image point of the infinite point can be detected by using the above method.So we now have at least three points which are the image of three infinite points.Using these image points,the vanishing line can be fitted.
After getting the vanishing line,compute the intersections of the vanishing line and the image of a circle,so we can get two points of the absolute conic.In this way,if we take photos of the model plane from at least three different angles,there will be enough points for recovering the absolute conic which has the degree of freedom of 5.Now we get the image of the absolute conic,and the intrinsic parameters of the camera.One fact we should point out is that the position of the camera is nothing to do with the image of absolute conic.That to say when we are taking photos from different angles,the image of absolute conic is unchanged.
In fact,proposition 0.1 has introduced a criterion of judging the relationship between two circles just by using the images of this two circles.
The motivation to study the geometry if conics arises from the fact that conics have more geometric information,and can be more robustly than points and lines.The the camera calibration method proposed in this paper,which based on the arbitrary distributed three(or more) coplanar circles,can deal with all the distributions of each two circles such as intersection,tangent, including,separation.What's more,we shouldn't know any of the properties of circles,and needn't handling matching.
As the absolute conic has all the information needed for the camera's intrinsic parameters,we now just need to find the image of the absolute conic. Based on this,many people have proposed some calibration methods.[27][2] give the calibration using a circle and its diameters;the linear method introduced by[6]based on two rectangulars;which the calibration algorithms in [7][14]based are only circles,but the method is not linear.
Aimed at computing the image of the absolute conic,which is also a conic,we should find at least five points which lay in the conic.Fortunately, the generally called circles p0ints are just such points.So the problem is now that how to find the images of at least three pairs of circle points.Realizing that circle points are the intersection of a circle and the infinite line,our task comes to finding the intersection of the images of a circle and the infinite line which is called the vanishing line.We can easily get the image of a circle which lav on the model plane,the coming difficulty is how to find the vanishing line.
Suppose there are three circles which are distributed arbitrarily,we will show how to find the vanishing line by using these there circles.We should at first group them so that two circles are in a group,then the image of at least one infinite point(which lay in the infinite line) will be computed from the image of each group.So using the there image points lay in the vanishing line,we can be fitted(of course we can use more than three circles in the model plane so as to get more image points of the vanishing line to fit it).
Suppose there is a point X=(x,y,t)~T in the projective plane of the circles O_1,O_2,the necessary and sufficient condition of X being the common pole is that O_2X=λO_1X,that is, (O_2-λO_1)X=0,(0-2-1) withλ∈R\{0}.
We have pointed out the fact that:the common poles of O_1,O_2 must include at least one infinite point.
Under the projective transformation H,circles O_1,O_2 have the image of quadratic curves C_1=λ_1H~(-T)O_1H~(-1),C_2=λ_2H~(-T)O_2H~(-1),withλ_1,λ_2 R\{0}.Substitute this into(0-2-1),we have:(C_2-λ(λ_2/λ_1)C_1)HX=0,that is, (C_2-λ′C_1)HX=0.(0-2-2) Therefore,HX is the common pole of C_1 and C_2.Because H is nonsingular, andλ_1,λ_2 are non-zero,so equation(0-2-1) and(0-2-2) are equivalent. This is just the algebra description of the invariance of pole and polar line about projective transformation.
Therefore,the common poles of C_1,C_2 which are the images of the two circle must include at least one infinite point.
The following is the method of finding the vanishing line base on the equation(0-2-3). det(C_2-λ′C_1)=0.(0-2-3)
proposition 0.1
(1) If the equation(0-2-3) haven't repeated root(in this situation the preimage of C_1,C_2 are neither concentric nor tangent circles),compute the independent real rootλ′,then substituteλ′into equation(0-2-2),the eigenvector HX is the image of infinite point d_(m∞).
(2) If equation(0-2-3) has repeated root,then it must be twofold root,compute this twofold rootλ′and substitute it into the matrix C_2-λ′C_1:
(a) If the matrix C_2-λ′C_1 rank of 1(showing that the preimage of C_1,C_2 are concentric circles),substitute thisλ′into equation(0-2-2),the two independent eigenvectors HX~((1)) and HX~2 are the image of infinite points d_(m∞)~((1)) and d_(m∞)~((2));
(b) If the matrix C_2-λ′C_1 rank of 2(showing that the preimage of C_1,C_2 are tangent circles),compute the independent real root of equation(0-2-3),then substitute it into equation(0-2-2),and the eigenvector HX is the image of infinite point d_(m∞).
Till now,we have find the image of a infinite point.
If there are three circles in the model plane,they can be divided into three groups,and at least one image point of the infinite point can be detected by using the above method.So we now have at least three points which are the image of three infinite points.Using these image points,the vanishing line can be fitted.
After getting the vanishing line,compute the intersections of the vanishing line and the image of a circle,so we can get two points of the absolute conic.In this way,if we take photos of the model plane from at least three different angles,there will be enough points for recovering the absolute conic which has the degree of freedom of 5.Now we get the image of the absolute conic,and the intrinsic parameters of the camera.One fact we should point out is that the position of the camera is nothing to do with the image of absolute conic.That to say when we are taking photos from different angles,the image of absolute conic is unchanged.
In fact,proposition 0.1 has introduced a criterion of judging the relationship between two circles just by using the images of this two circles.
引文
[1]马颂德,张正友.计算机视觉——计算理论与算法基础.科学出版社,北京,1998.
[2]胡培成,黎宁,周建江.一种改进的基于圆环点的摄像机自标定方法.光电工程,34(12):54-60,2007.
[3]范盛金.一元三次方程的新求根公式与新判别法.海南师范学院学报(自然科学版),2(2):91-98,12 1989.
[4]杨长江,孙凤梅,胡占义.基于平面二次曲线的摄像机标定.计算机学报,23(5):541-547,2000.
[5]吴福朝,胡占义.由二次曲线确定摄像机方位的线性算法.计算机学报,25(11):1157-1164,2002.
[6]吴福朝,王光辉,胡占义.由矩形确定摄像机内参数与位置的线性方法.软件学报,14(3):703-712,2003.
[7]胡钊政,谈正.利用二次曲线拟合和圆环点进行摄像机标定.西安交通大学学报,40(10):1065-1068,2006.
[8]蔡宇.3D重建中的摄像机标定与点云处理.硕士,吉林大学,吉林长春,5 2009.
[9]Hartlcy R,Zisscrman A.Multiple view geometry in computer vision.Cambridge University Press,2000.
[10]Liebowitz D.,Zisserman A.Metric rectification for perspective images of planes.In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition.Santa Barbara,page 482 488,CA,1998.IEEE Computer Societ.
[11]Penna M A.Determining camera parameters from the perspective projection of a quadrilateral.Pattern Recognition,24(6):533-541,1991.
[12]Semple J.G,Kneebone G.T.Algebraic Projective Geometry.Clarendon Press,Oxford,1952.
[13]Hui Zeng,Xiaoming Deng,Zhanyi Hu.A new normalized method on line-based homography estimation.Pattern Recognition Letters,29:1236-1244,2008.
[14]Yihong Wu,Xinju Li,Fuchao Wu,Zhanyi Hu.Coplanar circles,quasiaffine invariance and calibration.Image and Vision Computing,24:319-326,2006.
[15]Kanatabi,K.Geometric Computation for Machine Vision.Oxford Science Publications,Oxford,1993.
[16]J.-S.Kim,P.Gurdjos,I.-S.Kweon.Geometric and algebraic constraints of projected concentric circles and their applications to camera calibration.PAMI,27(4):637-642,2005.
[17]Wright J.,Wagner A.,Shankar Rao,Yi Ma.Homography from coplanar ellipses with application to forensic blood splatter reconstruction.Computer Vision and Pattern Recognition,pages 1250-1257,2006.
[18]Maybank SJ,Faugeras OD.A theory of self-calibration of a moving camera.International Journal of Computer Vision,8(2):123-151,1992.
[19]P.Sturm.Critical motion sequences for monocular self-calibration and uncalibrated euclidean reconstruction.Proc.CVPR' 97,pages 1100-1105,1997.
[20]Hartley R.Estimation of relative camera positions for uncalibrated cameras.Proceedings of the European Conference on Computer Vision,(NLCS 588):379-387,1992.
[21]Tsai R.An efficient and accurate camera calibration technique for 3d machine vision.Proceeding of Computer Vision and Pattern Recognition,pages 364-374,1986.
[22]Y.Ma,Stefano Soatto,Jana Kosecka,Shankar Sastry.Euclidean reconstruction and reprojection up to subgroups.Proc.ICCV'99,I:773-780,1999.
[23]Golub G,Loan C van.Matrix Computation.The John Hopkins University,Baltimore,Maryland,third edition,1996.
[24]Pierre Gurdjos,Peter Sturm,Yihong Wu.Euclidean structure from n ≥2parallel circles:Theory and algorithms.Computer Vision-ECCV 2006,3951:238-252,2006.
[25]Xianghua Ying and Hongbin Zha.Camera calibration using principal-axes aligned conics.ACCV 2007,Part Ⅰ,LNCS 4843,,pages 138-148,2007.
[26]Zhang Z.A flexible new technique for camera calibration.IEEE Transactions on Pattern Analysis and Machine Intelligence,22(11):1330-1334,2000.
[27]Meng XQ,Li H,Hu ZY.A new easy camera calibration technique based on circular points.In Thomas B Mirmehdi M,editor,Proceedings of the British Machine Vision Conference,pages 496-501,Bristol,2000.ILES Central Press.
[2]胡培成,黎宁,周建江.一种改进的基于圆环点的摄像机自标定方法.光电工程,34(12):54-60,2007.
[3]范盛金.一元三次方程的新求根公式与新判别法.海南师范学院学报(自然科学版),2(2):91-98,12 1989.
[4]杨长江,孙凤梅,胡占义.基于平面二次曲线的摄像机标定.计算机学报,23(5):541-547,2000.
[5]吴福朝,胡占义.由二次曲线确定摄像机方位的线性算法.计算机学报,25(11):1157-1164,2002.
[6]吴福朝,王光辉,胡占义.由矩形确定摄像机内参数与位置的线性方法.软件学报,14(3):703-712,2003.
[7]胡钊政,谈正.利用二次曲线拟合和圆环点进行摄像机标定.西安交通大学学报,40(10):1065-1068,2006.
[8]蔡宇.3D重建中的摄像机标定与点云处理.硕士,吉林大学,吉林长春,5 2009.
[9]Hartlcy R,Zisscrman A.Multiple view geometry in computer vision.Cambridge University Press,2000.
[10]Liebowitz D.,Zisserman A.Metric rectification for perspective images of planes.In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition.Santa Barbara,page 482 488,CA,1998.IEEE Computer Societ.
[11]Penna M A.Determining camera parameters from the perspective projection of a quadrilateral.Pattern Recognition,24(6):533-541,1991.
[12]Semple J.G,Kneebone G.T.Algebraic Projective Geometry.Clarendon Press,Oxford,1952.
[13]Hui Zeng,Xiaoming Deng,Zhanyi Hu.A new normalized method on line-based homography estimation.Pattern Recognition Letters,29:1236-1244,2008.
[14]Yihong Wu,Xinju Li,Fuchao Wu,Zhanyi Hu.Coplanar circles,quasiaffine invariance and calibration.Image and Vision Computing,24:319-326,2006.
[15]Kanatabi,K.Geometric Computation for Machine Vision.Oxford Science Publications,Oxford,1993.
[16]J.-S.Kim,P.Gurdjos,I.-S.Kweon.Geometric and algebraic constraints of projected concentric circles and their applications to camera calibration.PAMI,27(4):637-642,2005.
[17]Wright J.,Wagner A.,Shankar Rao,Yi Ma.Homography from coplanar ellipses with application to forensic blood splatter reconstruction.Computer Vision and Pattern Recognition,pages 1250-1257,2006.
[18]Maybank SJ,Faugeras OD.A theory of self-calibration of a moving camera.International Journal of Computer Vision,8(2):123-151,1992.
[19]P.Sturm.Critical motion sequences for monocular self-calibration and uncalibrated euclidean reconstruction.Proc.CVPR' 97,pages 1100-1105,1997.
[20]Hartley R.Estimation of relative camera positions for uncalibrated cameras.Proceedings of the European Conference on Computer Vision,(NLCS 588):379-387,1992.
[21]Tsai R.An efficient and accurate camera calibration technique for 3d machine vision.Proceeding of Computer Vision and Pattern Recognition,pages 364-374,1986.
[22]Y.Ma,Stefano Soatto,Jana Kosecka,Shankar Sastry.Euclidean reconstruction and reprojection up to subgroups.Proc.ICCV'99,I:773-780,1999.
[23]Golub G,Loan C van.Matrix Computation.The John Hopkins University,Baltimore,Maryland,third edition,1996.
[24]Pierre Gurdjos,Peter Sturm,Yihong Wu.Euclidean structure from n ≥2parallel circles:Theory and algorithms.Computer Vision-ECCV 2006,3951:238-252,2006.
[25]Xianghua Ying and Hongbin Zha.Camera calibration using principal-axes aligned conics.ACCV 2007,Part Ⅰ,LNCS 4843,,pages 138-148,2007.
[26]Zhang Z.A flexible new technique for camera calibration.IEEE Transactions on Pattern Analysis and Machine Intelligence,22(11):1330-1334,2000.
[27]Meng XQ,Li H,Hu ZY.A new easy camera calibration technique based on circular points.In Thomas B Mirmehdi M,editor,Proceedings of the British Machine Vision Conference,pages 496-501,Bristol,2000.ILES Central Press.